Hydrodynamic Traffic Flow Models and Its Application to Studying Traffic Control Effectiveness

Nickolay Smirnov, Alexey Kiselev, Valeriy WSEAS TRANSACTIONS on FLUID MECHANICS Nikitin, Mikhail Silnikov, Anna Manenkova

Hydrodynamic traffic flow models and its application to studying traffic control effectiveness

Nickolay Smirnov1,2, Alexey Kiselev 2, Valeriy Nikitin1,2, Mikhail Silnikov3, Anna Manenkova2

1 Scientific Research Institute for System Studies of the Russian Academy of Sciences, 36-1 Nakhimovskiy pr., Moscow 117218, RUSSIA 2Moscow M.V. Lomonosov State University, Leninskie Gory, 1, Moscow 119992, RUSSIA, 3 Saint Petersburg State Polytechnical University, 29 Politechnicheskaya Str., 195251 St. Petersburg RUSSIA

[email protected]

Abstract:-The present research was motivated by the arising difficulties in rational traffic organization the Moscow Government was faced in the recent years. The decrease of the roads handling capacity in the centre of the city (the Boulevard ring, the Gardens ring) caused by the increase of traffic density brought to necessity of re-organizing the traffic flows and constructing of the third ring road around the centre of the city. The present paper contains the results of investigations on mathematical modeling of essentially unsteady-state traffic flows on ring roads. The developed mathematical model is not limited only by the substance conservation equation and empirical relationships between density and velocity, but contains both the continuity (substance conservation) and momentum equations. It presents a closed form mathematical model and suggests the procedure to develop the model parameters based on technical characteristics of the vehicles. Thus, the available experimental data on the traffic flows observations is used only for validation of the model.

Key-Words:- Traffic, flow, model, continuous, handling capacity, traffic jam, regulation strategy

line of standing cars near the traffic light begins to 1 Introduction grow. Cars approaching from behind come to a stop. The rate of its growth is determined by the First attempts of developing the mathematical mean traffic flux and density on the road. It is, models for traffic flows were undertaken by actually, the velocity of a compression wave Lighthill and Whitham [1], Richards [2], described in [13-18]. The growth of the standing Greenberg [3], Prigogine [4]. A detailed analysis lane is limited by the time of approaching of the of the results obtained could be found in the book rarefaction wave propagating from the traffic light, by Whitham [5]. The authors regarded the traffic when it becomes green. The first characteristics of flow from the position of continuum mechanics this wave moves at a speed of k - the speed of applying empirical relationships between the flow weak disturbances. Then cars start to move slowly density and velocity of vehicles. Development of and the fast cars approaching from behind need to computer technique gave birth to continua models slow down to the low velocity of the traffic flow. application not only for analytical solutions [6] but That gives birth to a traffic jam, which moves also in numerical simulations of traffic flows [7- counter flow. One border of the jam moves at a 10]. Another approach using discrete speed of compression wave, the other is an approximation for public traffic and passengers expanding rarefaction wave. The size of the jam interaction is illustrated by [11-12]. gradually decreases. Depending on the distance of The presence of traffic regulation in the the traffic regulating section from the tunnel, the roads affect essentially flow dynamics. In standing line or traffic jam propagating counter particular, on changing the traffic light for red the flow could enter the tunnel. As it was shown, cars

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−1 0 0 ⎧−k lnρ , v< vmax , ρ =+1v/X l , V ()ρ = ⎨ *m( ()ax) v,00 vv≥ . ⎩ max max kX=+vln101− v 0 /l . The velocity V ()ρ is determined from the max ( ()max ) 0 dependence of the traffic velocity on density in the At vmax = 80 kph the deceleration way of “plane wave” when the traffic is starting from the an automobile of Lada brand is 45 m, which gives initial conditions ρ0 =1 and v0= , with account k = 35 kph when the mean length of a vehicle is of velocity limitation from above 0 . The 5 m (including the minimal distance between ()vv≤ max motionless vehicles). For this velocity v0 , the value of τ could be different for the cases of max acceleration or deceleration to the safe velocity maximal safe density is ρ* = 0.1. The maximal V ()ρ : accelerations of an automobile of this type are 2 2 + + − ⎪⎧τρ, V() < v a = 1.63 m/s and a = 5.5 m/s . τ = ⎨ − The “sound velocity” k estimated as above ⎩⎪τρ, V() ≥ v fits the experimental data [2, 3, 5] rather good. The remaining parameters in (2) are the Summarising the above, one comes to the following. YYLx=min{ 0 , −} is the model consisting of two PDE’s (1) and (2) characteristic visibility ahead depending on the describing the automobile traffic dynamics on a weather, ω ()y is the “weight” (in the single-lane road. The divergent form of those equations is: mathematical sense) of the traffic ahead of the ∂ρ ∂()ρv vehicle, which influences on the drivers decision +=0, of changing the velocity and which could be ∂∂tx determined as follows, for example: 2 ∂ ()ρv ∂ ()ρv ω0 ()y ⎧1, 0 ≤≤yY0 +=ρa , ωω()y ==Y , 0 ⎨ , ∂∂tx 0, yyY<> 0, ω y dy ⎩ 0 and the acceleration a is determined by three last ∫ 0 () expressions in (2). 0 We should consider the following boundary σ 0 is a dimensionless parameter ()01≤≤σ 0 conditions on the endings of the motorway section characterising the “weight” of local situation compared to the situation ahead. 0 ≤≤xL. Two variants of the conditions at the inflow x = 0 are possible: Thus, in the expression for the traffic 1) No traffic jam: the flow density is acceleration (right hand side of (2)) the first term given, and the velocity is equal to the maximal safe corresponds to the local situation influence, the for this density: second – to the influence of the traffic situation ρρ0,t = ; v0,t = vρ . ahead at the distance less or equal to Y , and the ( ) 0 ( ) ()0 third term is responsible to the tendency to 2) Traffic jam or jam at the inflow ending: maintain the velocity close to the maximal safe zero density gradient is assumed, and the velocity velocity V ()ρ . is equal to the maximal safe velocity at the density at the inflow: The estimate of the small disturbances ∂ρ propagation velocity k was made in [15-17] using = 0; v0,()()tV= ρ . x the following considerations. Let the traffic flow ∂ x=0 Presence or absence of the jam at the inflow start from a stationary state and (v0,== ρ 1) boundary of the domain is determined each time 0 after a time step calculation using the following accelerate until it reaches the velocity vmax at the criterion. The jam takes place, if two conditions maximal density ρ which still allows the safe * are true at x = 0: driving. We will denote the safe density as the maximal density at which the distance between ∂ρ > 0 and ρρ> 0 . vehicles is equal or more than the deceleration way ∂x x=0 length X ()v . Then, we get the following The “free outflow” condition is set on the expressions: outflow ending of the domain at xL= : ∂ρ ∂v = 0; = 0. ∂x ∂x

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As for the initial conditions, we assume that 3. In the moment when the street lights change to green, the maximal permitted the domain portion at 0 ≤≤xx0 is occupied with 0 velocity change back to vmax like in the other vehicles with density ρ0 moving at the velocity portions of the domain. , and the rest part is free from V ()ρ0 xxL0 <≤ the vehicles ()ρ ==0, v 0 . 3.2 ”Laying policemen”

3 Modelling traffic control devices This system of velocity limitation is modelled by maximal permitted velocity the We consider two most popular devices used position of the device which is much lower than to control automobile traffic on the urban roads: 0 one for the other portions of the road vmax . The street traffic lights and “laying policemen”. The present work deals with a system of two last is a set of low jumps across the road surface “policemen” laying at a given distance between which are placed at some distance from each other them; this case is mostly used in practice. The and force the drivers to decrease velocity position of the first device is denoted L . Then, the essentially in the zone of their placement. The 1 “laying policemen” are often used to provide maximal permitted velocity along the domain friendly conditions for pedestrians crossing roads 0 ≤≤xL is given as follows: in the vicinity of schools, institutions, ⎪⎧ v,p1 xLLd∈{} ,1+ , marketplaces, etc. v = max ⎨v,0 xLLLd0, , , ⎩⎪ max ∈[] {}1 1 + 3.1 Street traffic lights. 0 where vvpmax< , and the parameter v p (maximal velocity of passage) is one of the The main parameters of this device are the governing parameters. green, yellow and red signals duration, denoted tttgyr, , correspondingly. The following 4 Results of numerical investigations algorithm is proposed to model the street lights of traffic regulation strategies work. 1. In the moment when the signal changes to The numerical calculations of the problems yellow, the following distance is calculated: were processed using the TVD method with 2 second order of approximations (Van Leer’s flux v0 ()max limiter). The mesh had N = 201 grid nodes. xr = , 2ar The following parameters values were used where ar is the usual deceleration less than the in simulations: L = 1000 m is the length of the − deceleration of the emergency braking a . The domain; x0 = 100 m is the length of the domain vehicles which are closer to the street lights at this portion with vehicles at the first instance; moment than this distance, cannot stop and L = 500 m is the position of the traffic control therefore they pass the street lights, this yields the 1 devices (either street lights or the first “laying traffic rules. policemen”); d 50 m is the distance between 2. When the yellow signal is on, the maximal = the “laying policemen”; ρ0 =÷0.1 0.5 is the permitted velocity at the point xtl () is set to be 0 the following: inflow density, vmax = 25 m/s is the maximal t permitted velocity on the main portion of the road; vve x =0 ⋅ess , max ()l max v= 3 m/s is the maximal velocity of the te p “laying policemen” passage; k = 7.9 m/s is the where tess is the time left for the yellow signal to velocity of small disturbances propagation in the be on; xt() moves towards the street lights like: l automobile traffic flow; a+ = 1.5 m/s2 is the tess − 2 xLxl =1 −⋅r , maximal traffic flow acceleration; a = 5 m/s is ty the maximal (emergency) deceleration of the flow; 2 and L1 is the co-ordinate of the street lights ar = 1.5 m/s is the standard deceleration; device. As the result, by the moment of red lights Y0 = 100 m is the characteristic visibility ahead; all the traffic flow is blocked at the street lights. σ 0 = 0.7 is the local situation “weight”;

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τ + = 3.3 s , τ − =∞ are the delay times deviate is equal to 0.011629, and the maximal responsible to maintaining the safe velocity; difference is 0,021566. The figs. 3, 4 illustrate the profiles of ρ()x tg =÷40 300 s , ty = 5 s , tr = 30 s are for different time moments in the case of “laying durations of the street lights cycle. policemen”. The fig. 3 correspond to the inflow Thus, the varying parameters were the density ρ0 = 0.1, and the fig. 4 – to ρ0 = 0.3. inflow density ρ0 (together with the inflow The case ρ0 = 0.1 corresponds to the free velocity) and the green lights duration tg . traffic passage via the zone regulated by two The results are presented at the figs. 1 – 4 “laying policemen”, and the case ρ0 = 0.3 and in the Table 1. corresponds to origination of the traffic jam and its The figs. 1 and 2 illustrate traffic flow propagation counter-flow. It can be seen from both density distribution for the different time moments figs. 3 and 4 that two locations with increased which are shown on the figs, for the case of street density take place. In case ρ0 < 0.2 this does not lights traffic controlling. The green lights duration prevent the traffic flow from the free passage. If was taken tg = 50 s , and the inflow density was the inflow density is higher than 0.2, then the ρ = 0.18 (fig. 1) and ρ = 0.3 (fig. 2). The traffic jam originates in front of the first “laying 0 0 policeman”; it propagates towards the inflow time moments on those figures correspond to the portion and finally the density at the inflow following street lights cycles: fig. 1a – first cycle, increases thus decreasing the inflow velocity, and end of the green period; fig. 1b – first cycle, the traffic turns to be very slow (fig. 4). yellow period; fig. 1c – first cycle, end of the red period; fig. 1d – second cycle, green period; fig. 1e Green Critical Critical Table – second cycle, green period; fig. 1f – second lights density density 4.1. cycle, end of the red period. Fig. 2a corresponds to durati determined calculated Divergen the first cycle, yellow period; fig. 2b – first cycle, on, s numerically using formula ce end of the red period; fig. 2c – second cycle, end (4.3) of the green period; fig. 2d – fifth cycle, end of the 40 0,18 0,191679 0,011679 green period; fig. 2e – fifth cycle, end of the red 50 0,19 0,203729 0,013729 period; fig. 2f – sixth cycle, end of the green 60 0,21 0,213574 0,003574 period. It can be seen from the figures that at 70 0,22 0,221899 0,001899

ρ0 = 0.18 traffic jam does not occur (fig. 1), and 80 0,23 0,229109 -0,00089 90 0,23 0,23547 0,00547 at ρ = 0.3 the traffic jam arises and propagates 0 100 0,23 0,241159 0,011159 counter-flow decreasing vehicles velocity 110 0,25 0,246306 -0,00369 essentially. 120 0,25 0,251004 0,001004 The dependence of critical inflow density 130 0,26 0,255327 -0,00467 ∗ 140 0,26 0,259329 -0,00067 ρ0 under which the traffic jam does not occur, on the duration of the green period of the street 150 0,27 0,263054 -0,00695 160 0,27 0,266539 -0,00346 lights t was investigated, and the results are g 170 0,28 0,269813 -0,01019 placed in the Table 1. All the other parameters 180 0,28 0,2729 -0,0071 were fixed as shown above. The dependence of 190 0,29 0,275819 -0,01418 ∗ ρ0 on tg can be described by the following 200 0,29 0,278589 -0,01141 formula: 210 0,3 0,281224 -0,01878 220 0,3 0,283736 -0,01626 ρ∗ = abtln (3) 0 ()g 230 0,3 0,286136 -0,01386 where ab, are parameters depending on many 240 0,31 0,288434 -0,02157 250 0,31 0,290639 -0,01936 factors including the red lights duration tr . For the parameters used, we have got 260 0,31 0,292757 0,01724 a ==0,054, b 0,87 . The table 1 contains also 270 0,31 0,294795 -0,01521 280 0,31 0,296758 -0,01324 the difference of the critical inflow density 290 0,31 0,298653 -0,01135 obtained by traffic flow simulation and the value obtained using the formula (4.3). The mean square 300 0,31 0,300484 -0,00952

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a e

b f Fig. 1. Traffic flow density in the zone traffic regulation by “lights”. Time in seconds correspond to second

cycle, initial traffic density ρ0 = 0,18: a - the end of “green period”, b - “yellow period”, c - “yellow period”, d - “green period”, e – end of “green period”, f - end of the “red period”,

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b f Fig. 2. Traffic flow density in the zone traffic regulation by “lights”. Time in seconds correspond to sixth cycle,

initial traffic density ρ0 = 0.3 , a - “yellow period”, b - end of the “red period”, c - green period, d - end of the green period, e - end of the red period, f - end of the “green period ”.

Fig. 3. Traffic flow density in the zone traffic regulation by a “laying policemen”. Time in seconds is provided

on top of the figure; inflow traffic density ρ0 = 0.1.

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The results show (Fig. 4) that for relatively high inflow density artificial hump (“laying policemen”) device cannot provide the necessary handling capacity and brings to formation of a growing traffic jam, while for the lower inflow density (Fig. 3) the flow field comes to a steady- state regime. Thus the device successfully serves its function of reducing the traffic speed at a definite place of the road. 5. Conclusions The results of investigations testify that in a case of low inflow density of the automobile traffic the “laying policemen” devices allow to maintain traffic control without disturbing the free traffic flow. However, the increase of the inflow density leads to origination of the traffic jam and its propagation counter-flow. The regulation by the traffic lights allows improving the road handling capacity essentially when the duration of the different parts of the traffic lights cycle is properly chosen. The model developed accounts for the main feature of the automobile traffic flows: their self- b organisation. It allows describeing both qualitatively and quantitatively, the conditions for maximal road capacity, the origination and propagation of traffic jams, the influence of different traffic control devices.

Acknowledgements:

Russian Foundation for Basic Research (RFBR Grant 13-01-12056) is acknowledged for financial support.

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